On Symmetrical Convex Polytopes and their Edge Resolvability

Abstract

The \textit{metric dimension} of a graph Γ = ( V , E ) , denoted by dim ( Γ ) , is the least cardinality of a set of vertices in Γ such that each vertex in Γ is determined uniquely by its vector of distances to the vertices of the chosen set. The topological distance between an edge ε = y z ∈ E and a vertex k ∈ V is defined as d ( ε , k ) = min { d ( z , k ) , d ( y , k ) } . A subset of vertices R Γ in V is called an edge resolving set for Γ if for each pair of different edges e 1 and e 2 in E , there is a vertex j ∈ R Γ such that d ( e 1 , j ) ≠ d ( e 2 , j ) . An edge resolving set with minimum cardinality is called the edge metric basis for Γ and this cardinality is the edge metric dimension of Γ , denoted by dim E ( Γ ) . In this article, we show that the cardinality of the minimum edge resolving set is three or four for two classes of convex polytopes ( S n and T n ) that exist in the literature.